TL, DR. A likely, possible cause is that it is grease and as the cable passes through, it re-seals itself. Try using cold water to flush so that it gets chunky. Warm water after, for the final flush. If you snake+ cold water, flush. Snake+cold water flush, repeat and it gets a lil better each time, that's it. Never tried this but let a lot of ice melt overnight and then start. Also, wet/dry vac and try sucking after "P" trap at the start next morning. Might get lucky but will reseal (if it is resealing grease) and give you a real target concentration of muck-ball.
=====================================================================
If it is a grease clog. All information given would seem to suggest that. Jetting (mini) Might be an option. But second floor, residential. But a good professional knows how to deal with that.
What leads you to believe it is 100+ feet away? Is it alone utility sink or something like that? Is the kitchen residential? Disposal or a no longer used dishwasher? Are there no other fixtures on the same drain line? That is a very, very long line to do with a cable that is around 1/4 inch. The danger is it getting into a larger line and becoming a knot. I would look for a clean-out or fixture down the line. Which are a worry and a major headache.
Grease can sometimes reseal. Especially if hot water is used to flush it out after. It is counterintuitive to use cold water. But makes it chunky-er.
What size cable is the pipe and cables you used? The hand-driven, crank manor snakes are a lot more prone to boring a hole rather than jumping around enough to knock off all the muck, mire, and goop. Before the next attempt at using
Just for future reference. The chemical solution is bad for pipes and the environment. Doubly hard on pipes when it fails and sits there eating the pipe. Breplum is right. Getting that stuff on you or worse, in your eyes is not fun! I have stopped asking people if they have used it. I mean, I would try that first too. So much cheaper and if it works, yay. (Still shortens the life of the pipe)
But gonna be OK. Some real talent and experience here who'll get it right for you.